We employ recent work on computational noise to obtain near-optimal finite difference estimates of the derivatives of a noisy function. Our analysis employs a stochastic model of the noise without assuming a specific form of distribution. We use this model to derive theoretical bounds for the errors in the difference estimates and obtain an easily computable difference parameter that is provably near-optimal. Numerical results closely resemble the theory and show that we obtain accurate derivative estimates even when the noisy function is deterministic.